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%I #10 Feb 12 2019 18:23:10
%S 1,1,2,9,28,117,547,2671,14258,81335,490525,3124166,20903467,
%T 146330944,1068670938,8119799153,64030850716,522964211565,
%U 4415538541339,38476720288240,345511658683542,3192911107457165,30327219314890373,295738960383147498,2957765009957920537,30309760321885910842,317966262017144158784
%N G.f.: Sum_{n>=0} x^n * ((1+x)^n + sqrt(2)*i)^n / (1 + sqrt(2)*i*x*(1+x)^n)^(n+1), where i^2 = -1.
%C Note that the generating function expands into a power series in x with only real integer coefficients.
%H Paul D. Hanna, <a href="/A323682/b323682.txt">Table of n, a(n) for n = 0..300</a>
%F G.f.: Sum_{n>=0} x^n * ((1+x)^n + sqrt(2)*i)^n / (1 + sqrt(2)*i*x*(1+x)^n)^(n+1).
%F G.f.: Sum_{n>=0} x^n * ((1+x)^n - sqrt(2)*i)^n / (1 - sqrt(2)*i*x*(1+x)^n)^(n+1).
%e G.f.: A(x) = 1 + x + 2*x^2 + 9*x^3 + 28*x^4 + 117*x^5 + 547*x^6 + 2671*x^7 + 14258*x^8 + 81335*x^9 + 490525*x^10 + 3124166*x^11 + 20903467*x^12 + ...
%e Let r = sqrt(2)*i, so that r^2 = -2, then
%e A(x) = 1/(1 + r*x) + x*((1+x) + r)/(1 + r*x*(1+x))^2 + x^2*((1+x)^2 + r)^2/(1 + r*x*(1+x)^2)^3 + x^3*((1+x)^3 + r)^3/(1 + r*x*(1+x)^3)^4 + x^4*((1+x)^4 + r)^4/(1 + r*x*(1+x)^4)^5 + x^5*((1+x)^5 + r)^5/(1 + r*x*(1+x)^5)^6 + ...
%e also,
%e A(x) = 1/(1 - r*x) + x*((1+x) - r)/(1 - r*x*(1+x))^2 + x^2*((1+x)^2 - r)^2/(1 - r*x*(1+x)^2)^3 + x^3*((1+x)^3 - r)^3/(1 - r*x*(1+x)^3)^4 + x^4*((1+x)^4 - r)^4/(1 - r*x*(1+x)^4)^5 + x^5*((1+x)^5 - r)^5/(1 - r*x*(1+x)^5)^6 + ...
%o (PARI) {a(n) = my(r = sqrt(2)*I, A = sum(m=0,n+1, x^m*((1+x +x*O(x^n))^m + r)^m/(1 + r*x*(1+x +x*O(x^n))^m)^(m+1) )); round(polcoeff(A,n))}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n) = my(r = sqrt(2)*I, A = sum(m=0,n+1, x^m*((1+x +x*O(x^n))^m - r)^m/(1 - r*x*(1+x +x*O(x^n))^m)^(m+1) )); round(polcoeff(A,n))}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A323681, A323683, A323684, A323685.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Feb 12 2019
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